AP PRECALCULUS — UNIT 3B · TRIGONOMETRIC & POLAR FUNCTIONS
3.12B — Equivalent Representations of Trigonometric Functions
Notes — Sum, Difference, and Double-Angle Identities
💡 Learning Objectives (3.12.A Part 2)
By the end of this lesson you will be able to:
- Apply sum and difference identities for sine, cosine, and tangent
- Apply double-angle identities for sine and cosine
- Use these identities to find exact values for non-standard angles
- Simplify and verify expressions using sum, difference, and double-angle identities
1. Sum and Difference Identities
💡 Sum and Difference Identities
- sin(α + β) = sin α cos β + cos α sin β
- sin(α − β) = sin α cos β − cos α sin β
- cos(α + β) = cos α cos β − sin α sin β
- cos(α − β) = cos α cos β + sin α sin β
- tan(α + β) = (tan α + tan β) / (1 − tan α tan β)
- tan(α − β) = (tan α − tan β) / (1 + tan α tan β)
⚠️ Common mistakes
- sin(α + β) is NOT sin α + sin β. Trig functions don't distribute.
- Watch the SIGNS in the cosine identity: it's ‘cos cos − sin sin’ for the SUM, ‘cos cos + sin sin’ for the difference. The signs flip from what you might expect.
2. Finding Exact Values for Non-Standard Angles
📘 Example — sin(75°) exactly
- Recognize 75° = 45° + 30° (both special)
- Use sin(α + β) = sin α cos β + cos α sin β
- sin(75°) = sin 45° cos 30° + cos 45° sin 30°
- = (√2/2)(√3/2) + (√2/2)(1/2)
- = √6/4 + √2/4 = (√6 + √2)/4
📘 Example — cos(15°) exactly
- 15° = 45° − 30°
- cos(α − β) = cos α cos β + sin α sin β
- cos(15°) = cos 45° cos 30° + sin 45° sin 30°
- = (√2/2)(√3/2) + (√2/2)(1/2)
- = (√6 + √2)/4
3. Double-Angle Identities
By plugging α = β into the sum identities, we get DOUBLE-ANGLE formulas:
💡 Double-Angle Identities
- sin(2θ) = 2 sin θ cos θ
- cos(2θ) = cos²θ − sin²θ
- = 1 − 2 sin²θ
- = 2 cos²θ − 1
- tan(2θ) = 2 tan θ / (1 − tan²θ)
The cosine double-angle identity has THREE equivalent forms — pick whichever helps your problem. The third form (2 cos²θ − 1) is ideal if you know cos θ; the second (1 − 2 sin²θ) is ideal if you know sin θ.
4. Worked Example — Double Angles
📘 Example — Find sin(2θ) and cos(2θ) given sin θ = 4/5, θ in QII
- Quadrant II ⇒ cos θ < 0. Use Pythagoras: cos²θ = 1 − 16/25 = 9/25, so cos θ = −3/5
- sin(2θ) = 2 sin θ cos θ = 2(4/5)(−3/5) = −24/25
- cos(2θ) = 1 − 2 sin²θ = 1 − 2(16/25) = −7/25
5. Verifying Identities with Sum/Double-Angle
📘 Example — Verify sin(2θ) / sin θ = 2 cos θ
- Start LEFT: sin(2θ) / sin θ
- Use double-angle: (2 sin θ cos θ) / sin θ
- Cancel sin θ: 2 cos θ ✓
📘 Example — Verify (1 − cos 2θ)/(sin 2θ) = tan θ
- Top: 1 − cos 2θ. Use cos(2θ) = 1 − 2 sin²θ: 1 − (1 − 2 sin²θ) = 2 sin²θ
- Bottom: sin 2θ = 2 sin θ cos θ
- Ratio: (2 sin²θ) / (2 sin θ cos θ) = sin θ / cos θ = tan θ ✓
6. Solving Equations Using These Identities
Identities can transform a hard trig equation into a simpler one:
📘 Example — Solve sin(2x) = sin(x) on [0, 2π)
- Apply double-angle: 2 sin x cos x = sin x
- Move to one side: 2 sin x cos x − sin x = 0
- Factor: sin x · (2 cos x − 1) = 0
- So sin x = 0 ⇒ x = 0, π. And cos x = 1/2 ⇒ x = π/3 or 5π/3
- Solutions: {0, π/3, π, 5π/3}
⚠️ Common mistake
Don't divide both sides by sin(x) — you'd lose the solutions where sin(x) = 0. Always FACTOR instead.
7. Summary
- Sum and difference identities split a single trig of (α ± β) into combinations of trigs of α and β
- Double-angle identities are the special case α = β
- Cosine has three double-angle forms; pick the one that uses what you already know
- Use these identities to find exact values, simplify, verify, and solve equations
- When solving, factor — don't divide — to avoid losing solutions